1. **Problem statement:** Ava is comparing two tilings: one with squares and one with regular hexagons. Each tile has an area of 140 cm². (i) Find the side length of the square tile. (ii) Find the side length $x$ of the regular hexagon tile. --- 2. **Formula for area of a square:** $$\text{Area} = s^2$$ where $s$ is the side length. 3. **Calculate side length of square:** Given area = 140 cm², $$s^2 = 140$$ Taking square root, $$s = \sqrt{140}$$ Calculate: $$s \approx 11.8322$$ Rounded to 1 decimal place: $$s \approx 11.8 \text{ cm}$$ --- 4. **Formula for area of a regular hexagon:** A regular hexagon can be divided into 6 equilateral triangles. Area of hexagon: $$A = \frac{3\sqrt{3}}{2} x^2$$ where $x$ is the side length. 5. **Calculate side length $x$ of hexagon:** Given area = 140 cm², $$\frac{3\sqrt{3}}{2} x^2 = 140$$ Divide both sides by $\frac{3\sqrt{3}}{2}$: $$x^2 = \frac{140}{\frac{3\sqrt{3}}{2}} = 140 \times \frac{2}{3\sqrt{3}}$$ Show cancellation: $$x^2 = 140 \times \frac{2}{3\sqrt{3}}$$ Calculate numeric value: $$x^2 \approx 140 \times \frac{2}{3 \times 1.732} = 140 \times \frac{2}{5.196} \approx 140 \times 0.3849 = 53.89$$ Take square root: $$x = \sqrt{53.89} \approx 7.34$$ Rounded to 1 decimal place: $$x \approx 7.3 \text{ cm}$$ --- **Final answers:** (i) Side length of square tile: **11.8 cm** (ii) Side length of hexagonal tile: **7.3 cm**